What are independent trials, and how do they relate to the definition of probability?
In Probability we use the term Bernoullian trials for independent trials.
A sequence of trials are said to be Bernoullian if
(i) there are only two outcomes for each trial ( say a success or a failure)
(ii) Consecutive trials in the sequence are independent
and (iii) probability of success in every trial is a constant (namely p)
It is clear that no trial depends on the previous one here.
Consider drawing of two cards consecutively from a well shuffled deck of cards. Consider the event that cards are both diamonds. If the draw is with replacement the trials are independent ( Bernoullian) . Otherwise they are dependent.
In the former case the probability is ( 1/4) x (1/4)
In the later it is( 1/4) x (12/51)